The Optimal Use of Government Purchases for Macroeconomic Stabilization

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1 The Optimal Use of Government Purchases for Macroeconomic Stabilization Pascal Michaillat and Emmanuel Saez August 28, 2015 Abstract This paper extends Samuelson s theory of optimal government purchases by accounting for the contribution of government purchases to macroeconomic stabilization. Using a matching model of the macroeconomy, we derive a sufficient-statistics formula for optimal government purchases. The formula implies that the deviation of optimal government purchases from the Samuelson level is proportional to the elasticity of substitution between government and personal consumption times the government-purchases multiplier times the deviation of the unemployment rate from its efficient level. Hence, with a positive multiplier, optimal government purchases are above the Samuelson level when unemployment is inefficiently high and below it when unemployment is inefficiently low. We calibrate the formula to US data. A first implication is that US government purchases are optimal with a small multiplier of 0.04; if the multiplier is larger, US government purchases are not countercyclical enough. Another implication is that optimal government purchases should increase during recessions. With a multiplier of 0.5 the optimal government purchases-output ratio increases from 16.6% to 20.0% when the unemployment rate rises from the US average of 5.9% to 9%. With multipliers higher than 0.5 the optimal ratio increases less because fewer government purchases are required to fill the unemployment gap: with a multiplier of 2 the optimal ratio only increases from 16.6% to 17.6%; this is the same increase as with a multiplier of Keywords: government purchases, business cycles, multiplier, unemployment, matching Pascal Michaillat: Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK; p.michaillat@lse.ac.uk; Emmanuel Saez: Department of Economics, University of California Berkeley, 530 Evans Hall, Berkeley, CA 94720, USA; saez@econ.berkeley.edu; http: //eml.berkeley.edu/ saez/. We thank George Akerlof, Dirk Krueger, Emmanuel Farhi, Mikhail Golosov, Yuriy Gorodnichenko, David Romer, Stefanie Stancheva, Aleh Tsyvinski, and Pierre-Olivier Weill for helpful discussions and comments. This work was supported by the Center for Equitable Growth at the University of California Berkeley, the British Academy, the Economic and Social Research Council [grant number ES/K008641/1], and the Institute for New Economic Thinking. 1

2 1. Introduction In the United States the Full Employment and Balanced Growth Act of 1978 imparts the responsibility of achieving full employment to the Board of Governors of the Federal Reserve System and to the government. In practice, however, it is the Federal Reserve that has been in charge of macroeconomic stabilization. This reliance on the Federal Reserve reflects the consensus among policymakers and academics that monetary policy is the policy most adapted to stabilization. But the stabilization achieved through monetary policy alone remains imperfect. Of course, at the zero lower bound on nominal interest rates, monetary policy is severely constrained that is what happens after 2009 in Figure 1. 1 But that is not all: as Figure 1 shows, monetary policy was not subject to the zero lower bound in the 1991 and 2001 recessions, yet stabilization was only partial. Furthermore, local economies in a monetary union countries in the eurozone or US States cannot use monetary policy and must rely on other tools for stabilization. In this paper, we explore how government purchases can be used to improve stabilization. To that end, we embed the standard theory of optimal government purchases, developed by Samuelson [1954], within a matching model of the macroeconomy. 2 Samuelson showed that in a competitive, efficient model, the optimal provision of government consumption satisfies a simple formula: the marginal rate of substitution between government and personal consumption equals the marginal rate of transformation between government and personal consumption one in our model. But a matching model is not necessarily efficient. Our model builds on the matching framework from Michaillat and Saez [2015a]. There is one matching market where households sell labor services to other households and the government. In equilibrium there is some unemployment: sellers are unable to sell all the labor services that they could produce. The unemployment rate may not be efficient: the unemployment rate is inefficiently low when too many resources are devoted to purchasing labor services, and it is inefficiently high when too much of the economy s productive capacity is idle. 1 Krugman [1998] and Eggertsson and Woodford [2003] explain how the effectiveness of monetary policy is restricted by the existence of a zero lower bound on nominal interest rates. 2 A small literature analyzes optimal government purchases in disequilibrium models [for example, Drèze, 1985; Mankiw and Weinzierl, 2011; Roberts, 1982]. Since our model of unemployment is smoother than the disequilibrium model (see the discussion in Michaillat and Saez [2015a]), it provides nondegenerate policy trade-offs that can be resolved with optimal formulas expressed in estimable statistics. 2

3 10% 8% Unemployment rate 6% 4% 2% Federal funds rate 0% Figure 1: Unemployment and Monetary Policy in the United States, Notes: The unemployment rate is the quarterly average of the seasonally adjusted monthly unemployment rate constructed by the Bureau of Labor Statistics (BLS) from the Current Population Survey (CPS). The federal funds rate is the quarterly average of the daily effective federal funds rate set by the Board of Governors of the Federal Reserve System. The shaded areas represent the recessions identified by the National Bureau of Economic Research (NBER). When the unemployment rate is inefficient and government purchases influence unemployment, government purchases have an effect on welfare that is unaccounted for in Samuelson s theory. Hence, our formula for optimal government purchases adds to the Samuelson formula a correction term that measures the effect of government purchases on welfare through their influence on unemployment. The formula implies that optimal government purchases are above the Samuelson level if and only if government purchases bring unemployment closer to its efficient level; this occurs if unemployment is inefficiently high and government purchases lower unemployment, or if unemployment is inefficiently low and government purchases raise unemployment. We express our formula for optimal government purchases in terms of estimable sufficient statistics. 3 By virtue of being expressed with sufficient statistics, the formula applies broadly, irrespective of the specification of the utility function, aggregate demand, and price mechanism. We derive our results with exogenous labor supply, lump-sum taxation, and a representative household. The formula we derive is robust to introducing heterogeneous households, endogenous labor supply, and distortionary income taxation, paralleling the robustness of the Samuelson formula in public economics [Kaplow, 1996]. The formula shows that the deviation of optimal government purchases from the Samuelson 3 Chetty [2009] discusses the advantages of using sufficient statistics. The new dynamic public finance literature has also recently strived to express optimal policy formulas with estimable statistics [Golosov and Tsyvinski, 2015]. 3

4 level is proportional to the elasticity of substitution between government and personal consumption times the government-purchases multiplier times the deviation of the unemployment rate from its efficient level. The elasticity of substitution determines how the marginal rate of substitution between government and personal consumption depends on the ratio between government and personal consumption. The multiplier determines the effect of government purchases on unemployment. The deviation of unemployment from its efficient level determines the effect of unemployment on welfare. By showing how optimal government purchases in recessions depend on the multiplier and the marginal social value of the purchases (measured by the elasticity of substitution), our formula contributes to the policy discussions about the design of stimulus packages. A voluminous empirical literature estimates multipliers to describe the effects of government purchases on output and other variables. 4 Even though the empirical literature abstracts from welfare considerations, stimulus advocates believe in large multipliers and argue that government purchases can help fill the output gap in recessions [Romer and Bernstein, 2009]; conversely, stimulus skeptics believe in small, even negative, multipliers and argue that more government purchases could be detrimental [Barro and Redlick, 2011]. Our formula proposes a formal connection between the estimates of the government-purchases multiplier and the welfare-maximizing policy. Stimulus skeptics also warn that additional government spending could be wasteful. Our formula shows how the marginal social value of government purchases, measured by the elasticity of substitution, influences the size of the welfare-maximizing policy. We calibrate the formula to US data and use it to address several policy questions. First, we find that actual US government purchases, which are mildly countercyclical, are optimal under a minuscule multiplier of If the actual multiplier is larger than 0.04, US government purchases are not countercyclical enough. Second, we find that the formula implies significant increases in government purchases during recessions, even for small multipliers. With a multiplier of 0.1 the optimal government purchasesoutput ratio increases from 16.6% to 18.0% when the unemployment rate rises from the US average 4 In the literature estimating multipliers on aggregate US data, a few representative studies include Rotemberg and Woodford [1992], Ramey and Shapiro [1998], Blanchard and Perotti [2002], Galí, Lopez-Salido and Valles [2007], Mountford and Uhlig [2009], Hall [2009], Auerbach and Gorodnichenko [2012], Barro and Redlick [2011], and Ramey [2011b]. See Ramey [2011a] and Parker [2011] for excellent surveys. 4

5 of 5.9% to a high level of 9%; with a multiplier of 0.5 the optimal ratio increases more, from 16.6% to 20.0%. With multipliers higher than 0.5, however, the optimal government purchases-output ratio increases less: with a multiplier of 2 the optimal ratio only increases from 16.6% to 17.6%; this is the same increase as with a multiplier of In fact, we prove that the relation between the multiplier and the increase of the optimal government purchases-output ratio for a given increase in unemployment rate has a hump shape. For small multipliers, the optimal amount of government purchases is determined by the crowding out of personal consumption by government consumption; a higher multiplier means less crowding out and thus higher optimal government purchases. For large multipliers, it is optimal to fill the unemployment gap; a higher multiplier means that fewer government purchases are required to fill this gap. Our analysis is carried out without fleshing out a complete model of how government purchases affect unemployment and output because this is not needed to derive our sufficient statistics formulas which apply to a broad range of models. As an illustration, we use the simple dynamic model with an aggregate demand coming out of utility for wealth developed in Michaillat and Saez [2015b] can be used to calibrate any government purchase multiplier. If the interest rate is not fully flexible, shocks will create inefficient fluctuations in unemployment and the government purchase multiplier will be positive. We use this model to show that our formulas are good approximations to fully optimal policies. 2. A Macroeconomic Model of Unemployment and Government Purchases This section proposes a macroeconomic model of unemployment and government purchases, building on the matching framework from Michaillat and Saez [2015a]. The model is dynamic and set in continuous time. The model is generic in that we do not place much structure on the utility function, aggregate demand, price mechanism, and tax system. The components of the model that we introduce are sufficient to define a feasible allocation and describe the mathematical structure of an equilibrium. These are the only elements on which the optimal policy analysis relies. By maintaining this level of generality, we are able to show in Section 3 that our sufficient-statistics formula for optimal government purchases applies to a broad range of models. We provide a specific model building on Michaillat and Saez [2015b] as an example in Section 5. 5

6 The economy consists of a government and a measure 1 of identical households. Households are self-employed, producing and selling services on a matching market. 5 Each household has a productive capacity normalized to 1; the productive capacity indicates the maximum amount of services that a household could sell at any point in time. At time t, the household sells C(t) services to other households and G(t) services to the government. The household s output is Y (t) = C(t) + G(t). The matching process prevents households from selling their entire capacity so Y (t) < 1. The services are sold through long-term relationships. The idle capacity of the household at time t therefore is 1 Y (t). Since some of the capacity of the household is idle, some household members are unemployed. The rate of unemployment, defined as the share of workers who are idle, is u(t) = 1 Y (t), where Y (t) is the aggregate output of services. To purchase labor services, households and government advertise v(t) vacancies at time t. The rate h at which new long-term relationships are formed is given by a Cobb-Douglas matching function: h(t) = ω (1 Y (t)) η v(t) 1 η, where 1 Y (t) is aggregate idle capacity, v(t) is aggregate number of vacancies, ω > 0 governs the efficacy of matching, and η (0,1) is the elasticity of the matching function with respect to idle capacity. 6 The market tightness x is defined by x(t) = v(t)/(1 Y (t)). The market tightness is the ratio of the two arguments in the matching function: aggregate vacancies and aggregate idle capacity. With constant returns to scale in matching, the tightness determines the rates at which sellers and buyers enter into new long-term trading relationships. At time t, each of the 1 Y (t) units of idle productive capacity is sold at rate f (x(t)) = h(t)/(1 Y (t)) = ω x(t) 1 η and each of the v(t) vacancies is filled at rate q(x(t)) = h(t)/v(t) = ω x(t) η. The selling rate f (x) is increasing in x and the buying rate q(x) is decreasing in x; hence, when the tightness is higher, it is easier to sell services but harder to buy them. 5 We assume that households cannot consume their own labor services. To simplify the analysis, we abstract from firms and assume that all production directly takes place within households. Michaillat and Saez [2015a] show how the model can be modified to include firms hiring workers on a labor market and selling their production on a product market. 6 The empirical evidence summarized by Petrongolo and Pissarides [2001] indicates that a Cobb-Douglas specification for the matching function fits the data well. 6

7 Long-term relationships separate at rate s > 0. Accordingly, output is a state variable with law of motion Ẏ (t) = f (x(t)) (1 Y (t)) s Y (t). The term f (x(t)) (1 Y (t)) is the number of new relationships formed at t. The term s Y (t) is the number of existing relationships separated at t. However, in practice, because the transitional dynamics of output are fast, output rapidly adjusts to its steady-state level where market flows are balanced. 7 Throughout the paper, we therefore simplify the analysis by modeling output as a jump variable equal to its steady-state value defined by f (x(t)) (1 Y(t)) = s Y(t). With this simplification, output becomes a function of market tightness defined by Y (x) = f (x) f (x) + s. (1) The function Y (x) is in [0,1], increasing on [0,+ ), with Y (0) = 0 and lim x + Y (x) = 1. By definition, output is directly related to the unemployment rate: Y = 1 u. Hence, the simplification also implies that the unemployment rate is function of market tightness defined by u(x) = s s + f (x). (2) The function u(x) is in [0,1], decreasing on [0,+ ), with u(0) = 1 and lim x + u(x) = 0. Intuitively, when the market tightness is higher, it is easier to sell services so output is higher and the unemployment rate is lower. The elasticity of Y (x) is (1 η) u(x) and that of u(x) is (1 η) (1 u(x)). Advertising vacancies is costly. Posting one vacancy costs ρ > 0 services per unit time. Hence, a total of ρ v(t) services are spent at time t on filling vacancies. These services represent the resources devoted by households and government to matching with appropriate providers of services. Since these resources devoted to matching do not enter households utility function, we define two concepts of consumption. We refer to the quantities C(t) and G(t) purchased by households and government as gross personal consumption and gross government consumption. Following common usage, government consumption designates the consumption by households of services purchased by the government. We define the net personal consumption c(t) < C(t) and net gov- 7 Hall [2005] and Shimer [2012] establish this property for the employment rate, which is proportional to output in our model. Appendix A confirms this result over a longer time period. 7

8 x m Market tightness x 0 0 Gross output: Net output: y(x) = Y (x) 1+ (x) Y (x) =1 u(x) Matching costs: y(x) (x) Unemployment rate: s u(x) = s + f(x) Labor services 1 x x* 0 0 y(x) y* Inefficiently low unemployment Efficient unemployment u* Inefficiently high unemployment 1 y A. Feasible allocations B. Efficient and inefficient unemployment rates Figure 2: The Market for Labor Services ernment consumption g(t) < G(t) as the gross consumptions net of the services used for matching. We also refer to Y (t) = C(t) + G(t) as gross output and y(t) = c(t) + g(t) as net output. As market flows are balanced, s Y (t) = v(t) q(x(t)). Hence, y(t) = Y (t) ρ v(t) = Y (t) ρ s Y (t)/q(x(t)), which implies that Y (t) = [1 + τ(x(t))] y(t) where we define τ(x) = ρ s q(x) ρ s. Of course we also have C(t) = [1 + τ(x(t))] c(t) and G(t) = [1 + τ(x(t))] g(t). Hence, enjoying one service requires to purchase 1 + τ services one service that enters the utility function plus τ services for matching. The matching wedge τ(x) is positive and increasing on [0,x m ), where x m (0,+ ) is defined by q(x m ) = ρ s. In addition, lim x x m τ(x) = +. Intuitively, when the market tightness is higher, it is more difficult to match with a seller so the matching wedge is higher. The elasticity of τ(x) is η (1 + τ(x)). The concepts of gross consumption and gross output correspond to the quantities measured in national accounts. 8 Indeed, gross output is proportional to employment in our model, and part of employment measured in national accounts is used to create matches for instance, human resource workers, placement agency workers, procurement workers, buyers even though the services they provide are used for matching and do not enter households utility. 8 In the US National Income and Product Accounts (NIPA), C(t) is personal consumption expenditures and G(t) government consumption expenditures. 8

9 It is useful to write net output as a function of market tightness: y(x) = 1 u(x) 1 + τ(x). (3) This function y(x) plays a central role in the analysis because it gives the amount of services that can be allocated between net personal consumption and net government consumption for a given tightness. The expression (3) shows that net consumption is below the productive capacity (normalized to 1) because some services are not sold in equilibrium (u(x) > 0) and some services are used for matching instead of net consumption (τ(x) > 0). The function y(x) is defined on [0,x m ], positive, with y(0) = 0 and y(x m ) = 0. The elasticity of y(x) is (1 η) u(x) η τ(x). Hence, the elasticity of y(x) is 1 η at x = 0, and it is at x = x m, and it is strictly decreasing in x. Therefore, the function y(x) is strictly increasing for x x, strictly decreasing for x x, where the tightness x is defined by (1 η) u(x ) = η τ(x ). (4) The function y(x) is therefore maximized at x = x. Since x maximizes net output, we refer to it as the efficient tightness. The efficient tightness is the tightness underlying the condition of Hosios [1990] for efficiency in a matching model. The efficient unemployment rate is u u(x ). Figure 2 summarizes the model. Panel A depicts how net output, gross output, and unemployment rate depend on market tightness in feasible allocations. Panel B depicts the function y(x), the efficient tightness x, the efficient unemployment rate u, and situations in which tightness is inefficiently high and unemployment is inefficiently low (x > x, u < u ), and situations in which tightness is inefficiently low and unemployment is inefficiently high (x < x, u > u ). When unemployment is inefficiently high, too much of the economy s productive capacity is idle, and a marginal decrease in unemployment increases net output. When unemployment is inefficiently low, too many resources are devoted to purchasing labor services, and a further decrease in unemployment reduces net output. 9 9 In our model gross output, Y, is proportional to the employment rate, 1 u; hence, when output is 1 percent below trend, the employment rate is 1 percent below trend and the unemployment rate is slightly less than 1 percentage point above trend. This property seemingly contradicts Okun s law; Okun [1963] found that in US data for , output was 3 percent below trend when the unemployment rate was 1 percentage point above trend. The relationship 9

10 Next, we assume that the government s budget is balanced at all times using a lump-sum tax T (t) = G(t) levied on households. 10 We also assume that the government sets g(t) as a function of the other variables at time t and parameters. In that case, the dynamical system describing the equilibrium of the model only has jump variables and no state variables. We assume that the equilibrium system is a source. 11 Hence, the equilibrium converges immediately to its steady-state value from any initial condition. Since transitional dynamics are immediate, an equilibrium is completely characterized by its steady state. Finally, we define a feasible allocation and an equilibrium. We give a static definition as the equilibrium converges immediately to its steady state: DEFINITION 1. A feasible allocation is a net personal consumption c [0, 1], a net government consumption, g [0, 1], and net output y [0, 1], and a market tightness x [0, + ) that satisfy y = y(x) and c = y g. The function y(x) is defined by (3). DEFINITION 2. An equilibrium function is a mapping from a net government consumption g to a feasible allocation [c,g,y,x]. Given that y and c are functions of x and g in a feasible allocation, the equilibrium function is summarized by a mapping from a net government consumption g to a tightness x. In the model, an equilibrium is just a value of the equilibrium function. In practice the equilibrium function x(g) arises from the household s optimal consumption choice and the price mechanism. The function x(g) can describe efficient prices, bargained prices, or rigid prices. To provide a concrete example, Section 5 describes the function x(g) in a specific model. between output gap and unemployment gap has evolved over time, however. In Appendix B, we estimate Okun s law for the entire period and for the recent period. We find that when the unemployment rate is 1 percentage point above trend, output is 1.8 percent below trend in the period and 1.3 percent below trend in the period. 10 When households are Ricardian in the sense that they do not view government bonds as net wealth because such bonds need to be repaid with taxes later on financing government purchases with debt is economically equivalent to maintaining budget balance using a lump-sum tax [Barro, 1974]. Hence, our analysis would remain valid if the government financed government purchases with debt and households were Ricardian. Michaillat and Saez [2015b] analyze debt policy when individuals are not necessarily Ricardian. 11 Without this assumption, the model would suffer from dynamic indeterminacy, making the welfare analysis impossible. The complete model we analyze in Section 5 satisfies the source system assumption. 10

11 3. Sufficient-Statistics Formulas for Optimal Government Purchases The representative household derives instantaneous utility U (c, g) from net personal consumption c and net government consumption g. The function U is increasing in its two arguments, concave, and homothetic. The marginal rate of substitution between government consumption and personal consumption is MRS gc U / g U / c. Since U is homothetic, the marginal rate of substitution is a decreasing function of g/c = G/C. 12 Since the equilibrium immediately converges to its steady state, the welfare of an equilibrium is U (c,g). In a feasible allocation, net personal consumption is given by c = y(x) g, so welfare can be written as U (y(x) g,g). Given an equilibrium function x(g), the government chooses g to maximize welfare U (y(x(g)) g,g). We assume that the welfare function g U (y(x(g)) g,g) is well behaved: it admits a unique extremum and the extremum is a maximum. 13 Under this assumption, first-order conditions are not only necessary but also sufficient to describe the optimum of the government s problem. In this section we derive several sufficient-statistics formulas giving the optimal level of government purchases. These formulas are adapted to answer different questions. The first formula relates the optimal level of government purchases to the level given by the Samuelson formula. The first formula is exact but it is not expressed with statistics that can be estimated in the data; the second and third formula are approximate but they are expressed with estimable statistics, which makes them appropriate for practical policy applications. The second formula relates the deviation of 12 By homothetic, we mean that the utility can be written as U (c,g) = W (w(c,g))) where the function W is increasing and the function w is increasing in its two arguments, concave, and homogeneous of degree 1. Since w is homogeneous of degree 1, its derivatives w/ c and w/ g are homogeneous of degree 0. Combining these properties, and we have MRS gc = w g (c,g) W (w(c,g))) w c (c,g) W (w(c,g))) = w g w c ( 1, g) c ( cg,1). We see that MRS gc is a function of g/c only. As w is concave, w/ g is decreasing in its second argument while w/ c is decreasing in its first argument; hence, MRS gc is a decreasing function of g/c. 13 We showed that x y(x) has a unique extremum and this extremum is a maximum. We assumed that U is concave. Therefore, we need g x(g) to be well behaved in order for g U (y(x(g)) g,g) to satisfy the assumption. 11

12 optimal government purchases from the Samuelson level to the government-purchases multiplier, the elasticity of substitution between government and personal consumption, and the deviation of actual market tightness from efficient market tightness. The second formula is helpful to assess actual government purchases, but it only defines optimal government purchases implicitly, so it does not say how government purchases should be adjusted after a macroeconomic shock. Thus, we propose a third formula that expresses optimal government purchases as an explicit function of the change in unemployment rate observed after the shock, the government-purchases multiplier, and the elasticity of substitution between government and personal consumption An Exact Formula Taking the first-order condition of the government s problem, we obtain 0 = U g U c + U c y (x) x (g). Reshuffling the terms in the optimality condition and dividing the condition by U / c yields the formula for optimal government purchases: PROPOSITION 1. Optimal government purchases satisfy 1 = MRS gc + y (x) x (g). (5) As is standard in optimal tax formulas, formula (5) characterizes the optimal level of government purchases implicitly. If the formula holds, then g maximizes welfare. If the right-hand side of (5) is above 1, a marginal increase in g raises welfare; conversely, if the right-hand side is below 1, a marginal increase in g reduces welfare. Although the formula is only implicit, it is useful because it transparently shows the economic forces at play. Formula (5) is the formula of Samuelson [1954] plus a correction term. The Samuelson formula is 1 = MRS gc ; it requires that the marginal utilities from personal consumption and government consumption are equal. With homothetic preferences, MRS gc decreases with G/C, so the Samuelson formula determines a unique ratio G/C denoted by (G/C). As G/Y = (G/C)/(G/C + 1), it also defines a unique government purchases-output ratio G/Y denoted by (G/Y ). The correction 12

13 Table 1: Optimal Government Purchases-Output Ratio Compared to Samuelson Ratio Effect of net government consumption on unemployment Unemployment rate du/dg > 0 du/dg = 0 du/dg < 0 Inefficiently high lower same higher Efficient same same same Inefficiently low higher same lower Notes: The government purchases-output ratio in the theory of Samuelson [1954] is given by 1 = MRS gc. Formula (5) implies that compared to the Samuelson ratio, the optimal government purchases-output ratio is higher if y (x) x (g) > 0, same if y (x) x (g) = 0, and lower if y (x) x (g) < 0. By definition, the unemployment rate is inefficiently high when y (x) > 0, inefficiently low when y (x) < 0, and efficient when y (x) = 0. Last, du/dg = u (x) x (g) where u(x) is given by (2). Since u (x) > 0, du/dg and x (g) have the same sign. term is the product of the effect of government purchases on tightness, x (g), and the effect of tightness on net output, y (x). The correction term measures dy/dg; it is positive if and only if more government purchases yield higher net output. Given the relation between net output, tightness, and unemployment rate (Figure 2), the correction term is positive when government purchases bring the unemployment rate toward its efficient level. Our formula gives general conditions for the optimal level of government purchases to be above or below the Samuelson level. The formula indicates that the government purchases-output ratio should be above the Samuelson ratio if the correction term is positive, and below the Samuelson ratio if the correction term is negative. If the correction term is zero, the optimal government purchases-output ratio satisfies the Samuelson formula. There are two situations in which the correction term is zero and the optimal government purchases-output ratio is given by the Samuelson formula. The first is when the unemployment rate is efficient (y (x) = 0). In that case, the marginal effect of government purchases on unemployment has no first-order effect on welfare and the principles of Samuelson s theory apply. The second is when government purchases have no effect on tightness and thus on the unemployment rate (x (g) = 0). In that case, the model is isomorphic to Samuelson s framework. In all other situations, the correction term is nonzero and the optimal government purchasesoutput ratio departs from the Samuelson ratio. The formula implies that the optimal government purchases-output ratio is above the Samuelson ratio if and only if government purchases bring 13

14 unemployment closer to its efficient level. This occurs either if the unemployment rate is inefficiently high and government purchases lower it, or if the unemployment rate is inefficiently low and government purchases raise it. Table 1 summarizes all the possibilities. The results described here are closely related to those obtained by Farhi and Werning [2012]. Farhi and Werning study the optimal use government purchases for stabilization in a fiscal union. 14 In their model, government purchases of nongraded goods increase output with a multiplier of 1. They find that optimal government purchases are given by the Samuelson formula plus a correction term equal to a labor wedge that measures of the state of the business cycle. Consistent with our analysis, they find that government purchases should be provided above the Samuelson level in recessions and below the Samuelson level in booms. The results are also consistent with those obtained by others in new Keynesian models. Woodford [2011] notes that away from the zero lower bound, monetary policy perfectly stabilizes the economy; hence, government purchases are not needed for stabilization, and they should follow the Samuelson formula. We obtain the same result: when unemployment is efficient, government purchases are given by the Samuelson formula. Werning [2012] describes the optimal use of government purchases in a liquidity trap. Like us, he finds that the optimal level of government purchases is the Samuelson level plus a correction term arising from stabilization motives. Furthermore, formula (5) can be used to recover the results on government purchases obtained in the Keynesian regime of disequilibrium models pioneered by Barro and Grossman [1971]. The correction term in (5) can be written as dy/dg. In a disequilibrium model, there are no matching costs so y = Y and g = G and the correction term is equal to the standard multiplier dy /dg. In the Keynesian regime, personal consumption is fixed because it is determined by aggregate demand and the above-market-clearing price; hence, there is no crowding out of personal consumption by government consumption and dy /dg = 1. On the other hand when the product market clears, crowding out is one-for-one and dy /dg = 0. We assume that there is some value for government purchases such that MRS gc > 0. Since MRS gc +dy /dg > 1 as long as the output gap is not closed, our formula implies that additional government purchases raise welfare in the Keynesian regime and that it is optimal to use government purchases to fill entirely the output gap. Finally, the structure of the formula a standard formula from public economics plus a cor- 14 See also Galí and Monacelli [2008] for an analysis of optimal monetary and fiscal policy in a currency union. 14

15 rection term capturing stabilization motives is similar to the structure of the formula for optimal unemployment insurance derived by Landais, Michaillat and Saez [2010] in a matching model, and to the structure of the formulas for optimal macroprudential policies derived by Farhi and Werning [2013] in models with price rigidities An Implicit Formula in Estimable Statistics Formula (5) is useful to describe the economic forces at play, but it is difficult to use it for practical policy applications because it is not expressed with estimable statistics. The correction term in the formula can be written as a multiplier dy/dg. But this multiplier is not directly estimable in the data because it gives the effect of net government consumption on net output whereas the data measure gross government consumption and gross output. To adapt formula (5) for policy applications, we re-express it with estimable statistics. The main task is to express dy/dg as a function of the government-purchases multiplier dy /dg estimated by macroeconomists in aggregate data, and other estimable statistics. We first obtain an exact formula in Lemma 1 and then provide a much simpler approximation in Proposition 2. LEMMA 1. Optimal government purchases satisfy ( 1 = MRS gc + 1 η 1 η τ(x) ) dy ( u(x) dg 1 η 1 η τ(x) u(x) G Y dy ) 1. (6) dg All the statistics are defined above and listed in Table 2. Proof. First, note that d ln(y) d ln(g) = d ln(y) d ln(x) d ln(x) d ln(g) d ln(g) d ln(g). Next, as the elasticity of Y (x) is (1 η) u, we find that d ln(x) d ln(g) = 1 (1 η) u d ln(y ) d ln(g). 15

16 Last, using G = (1 + τ(x)) g and as the elasticity of 1 + τ(x) is η τ, we find that d ln(g) d ln(g) = 1 + η τ d ln(x) d ln(g) d ln(g) d ln(g). Combining this equation with the expression for d ln(x)/d ln(g) obtained above, we get ( d ln(g) d ln(g) = 1 η 1 η τ u d ln(y ) ) 1. d ln(g) Combining all these results and as the elasticity of y(x) is (1 η) u η τ, we obtain ( dy dg = 1 η 1 η τ ) dy ( u dg 1 η 1 η τ u d ln(y ) ) 1. d ln(g) Bringing all the elements together, we obtain (6). Next, relying on first-order approximations, we obtain the following formula: PROPOSITION 2. Optimal government purchases approximately satisfy G/C (G/C) (G/C) ε m x x x, (7) where ε d ln(mrs gc) d ln(g/c) (8) is the elasticity of substitution between government and personal consumption, m dy /dg 1 (G/Y ) (dy /dg) (9) is an increasing function of the government-purchases multiplier dy /dg, and (G/C) and x are defined above and listed in Table 2. The statistics ε and m are evaluated at [(G/C),x ]. The approximation is valid up to a remainder that is O((x x ) 2 + (G/C (G/C) ) 2 ). The rigorous proof is presented in Appendix C but the heuristic derivation of (6) is simple. First, by definition of the marginal rate of substitution, we have MRS gc (G/C) MRS gc ((G/C) ) 16

17 Table 2: The Sufficient Statistics Used in the Formulas for Optimal Government Purchases Notation Description Definition u(x) unemployment rate u(x) = s f (x)+s τ(x) matching wedge τ(x) = s ρ 1 η elasticity of the selling rate 1 η = a decreasing function of 1 η a = y (x) marginal effect of tightness on net output d ln(y) d ln(x) q(x) s ρ d ln( f (x)) d ln(x) (G/Y ) (1 G/Y ) (1 η) u = (1 η) u(x) η τ(x) x efficient tightness y (x ) = 0 u efficient unemployment rate u = u(x ) MRS gc (G/C) (G/Y ) ε x (g) dy /dg marginal rate of substitution between government and personal consumption Samuelson ratio of government consumption to personal consumption Samuelson ratio of government consumption to output elasticity of substitution between government and personal consumption marginal effect of net government consumption on equilibrium tightness government-purchases multiplier m increasing function of dy /dg m = MRS gc = U / g U / c MRS gc ((G/C) ) = 1 (G/Y ) = (G/C) 1+(G/C) ε = d ln(mrs gc) d ln(g/c) dy /dg 1 (G/Y ) (dy /dg) ( 1/ε) (G/C (G/C) )/(G/C). As MRS gc ((G/C) ) = 1, we have 1 MRS gc (1/ε) (G/C (G/C) )/(G/C) so we only need to show that the last term in equation (6), which is a product of three terms, is approximately equal to m (x x )/x. Second, at x = x, from (4) we have (η/(1 η)) (τ/u) = 1. Hence, the product of the last two terms of the three term product in equation (6) is approximately m when x is close to x. (η/(1 η)) (τ/u) = 1 at x = x leads to the first-order expansion 1 (η/(1 η)) (τ(x)/u(x)) ζ (x x ) where ζ is the derivative of [η/(1 η)] τ(x)/u(x) with respect to x evaluated at x. We show in Appendix C that α = 1/x which is the only non obvious step that completes the proof. 15 Like formula (5), formula (7) is implicit because its right-hand-side is endogenous to the policy. 15 This derivative result is consistent with the result in Lemma 2 that [η/(1 η)] τ(x)/u(x) x/x for all x. 17

18 If the right-hand side of (7) is higher than the left-hand side, a marginal increase in G would raise welfare; conversely, if the right-hand side is lower than the left-hand side, a marginal increase in G would reduce welfare. Although the formula is only implicit, it is useful to assess actual policy, as showed in Section 4. Formula (7) shows that the elasticity of substitution between government and personal consumption is critical to determine the optimal level of government purchases. The elasticity plays an important role because it determines how quickly the marginal value of government purchases relative to that of personal consumption fades when government purchases increase. The role of this elasticity has been largely neglected in previous work. Consider the case with ε = This would be a situation in which we need a certain number of bridges for an economy of a given size, but beyond that number, additional bridges have zero value ( bridges to nowhere ). The formula says that with ε = 0 the ratio G/C should stay at the Samuelson ratio (G/C), irrespective of the level of unemployment. Increasing G/C beyond (G/C) is never optimal because government consumption g is useless at the margin for G/C beyond (G/C) and personal consumption c is always crowded out by g. 17 Consider next the case with ε This would be a situation in which the services provided by the government substitute exactly for the services purchased by individuals on the market. The formula says that with ε + government purchases should completely fill the tightness gap such that x = x, even if government purchases crowd out personal consumption. Government purchases should be used to maximize net output, or equivalently bring tightness to x, because only net output matters for welfare -the composition of output does not. In reality, government purchases probably have some value at the margin, without being perfect substitute for personal consumption; that is, ε > 0 but ε < +. In that case, the ratio G/C and tightness x generally departs from the Samuelson ratio (G/C) and from the efficient tightness x. Formula (7) also shows that the welfare-maximizing level of government purchases depends on the government-purchases multiplier, confirming an intuition that macroeconomists have had for 16 The Leontief utility function U (c,g) = min{(1 γ) c,γ g} has ε = 0. With this utility function, the Samuelson ratio is (G/C) (1 γ)/γ. 17 If g did not crowd out c, then dc/dg > 0 and dy/dg = 1 + dc/dg > 1, which would violate (5) and can therefore not occur at an optimum. 18 The linear utility function U (c,g) = c + g has ε = +. 18

19 a long time. 19 The multiplier dy /dg enters through the statistic m defined by (9) in the formula. The statistic m is an increasing function of dy /dg, m = 0 when dy /dg = 0, and m + when dy /dg Y /G. Clearly, m and dy /dg have the same sign. The statistic m enters the formula instead of dy /dg because what matters for welfare is dy /dg, not dy /dg, and at x, dy /dg = (1 + τ(x )) m > dy /dg. If the government-purchases multiplier is zero, government purchases should remain at the level given by the Samuelson formula. If the multiplier is positive, the government purchasesoutput ratio should be above the Samuelson ratio when unemployment is inefficiently high and below it when unemployment is inefficiently low. If the multiplier is negative, the government purchases-output ratio should be below the Samuelson ratio when unemployment is inefficiently high and above it when unemployment is inefficiently low. Formula (7) is a first-order approximation of (5) valid up to a remainder that is O((x x ) 2 + (G/C (G/C) ) 2 ). In practice, government purchases are never far from the Samuelson level so (G/C (G/C) ) 2 remains small. However, as we shall see in Section 4, tightness displays large fluctuates. Hence, it could sometimes be far from its efficient level, so (x x ) 2 could be large and formula (7) could be inaccurate. To alleviate this concern, we show that under some reasonable assumptions, a formula similar to (7) provides a good approximation of (5) even far from the efficient tightness. LEMMA 2. Assume that the separation rate s and matching cost ρ are small enough compared to the matching rates f (x) and q(x). Then the following is a good approximation: η 1 η τ(x) u(x) x x. (10) Proof. Assume that s f (x) and s ρ q(x). In that case, f (x) is a good approximation of s + f (x) and q(x) of q(x) s ρ. Therefore, we can approximate τ(x) u(x) = s ρ q(x) s ρ s + f (x) s ρ s q(x) f (x) = ρ x. s This approximation implies that τ(x )/u(x ) ρ x. The efficiency condition (4) implies that 19 For instance, using a heuristic method, Woodford [2011] and Nakamura and Steinsson [2014] show that the multiplier affects the optimal level of government purchases in a new Keynesian model. 19

20 τ(x )/u(x ) = (1 η)/η. Hence, (1 η)/η ρ x. Combining τ(x)/u(x) ρ x and η/(1 η) 1/(ρ x ), we find that (10) is a good approximation. Combining the results from Lemmas 1 and 2, we obtain the following formula: PROPOSITION 3. Assume that the separation rate s and matching cost ρ are small compared to the matching rates f (x) and q(x). Assume also that G/C remains in the neighborhood of (G/C). Then a good approximation for optimal government purchases is G/C (G/C) dy /dg (G/C) ε 1 (x/x ) (G/Y ) (dy /dg) x x x. (11) All the statistics are defined above and listed in Table 2. The statistics ε and dy /dg are evaluated at [G/C,x]. Proof. We start from (6). Using Lemma 2, we approximate 1 [η/(1 η)] (τ/u) by (x x )/x and [η/(1 η)] (τ/u) by x/x. Combining these approximations yields (11). An advantage of formula (11) is that it applies even when business cycles have triggered large departures of the tightness from its efficient level. While the formula relies on assumptions on the size of some parameters and variables (s and ρ small, gap between G/C and (G/C) small), we find that these assumptions are satisfied in US data, suggesting that (11) would be accurate in the US. Appendix A finds that in US monthly data s = 3.3%, s ρ = 6.5%, f (x ) = 56%, and q(x ) = 94%, validating the assumptions that s f (x) and s ρ q(x). Appendix A also finds that in US data for , the approximation of Lemma 2 is extremely accurate. Panel B of Figure 3 shows that in US data the deviation of G/C from (G/C) is never more than 8%, validating the assumption that the gap between G/C and (G/C) is small. To ascertain the conditions under which formula (7) holds far from the efficient tightness, it suffices to compare (7) and (11). A first condition is that the elasticity (G/Y ) (dy /dg) is small enough such that the large fluctuations of x/x in the denominator of the right-hand side of (11) do not generate large deviations of the right-hand side of (11) from the right-hand side of (7). This condition seems satisfied in US data. Section 4 shows that G/Y = 0.17 on average in US data and argues that a reasonable estimate of the multiplier is dy /dg = 0.6, suggesting that the elasticity 20

21 (G/Y ) (dy /dg) is fairly small. Appendix A also finds that 1 (x/x ) (G/Y ) (dy /dg) is always quite close to 1 (G/Y ) (dy /dg). A second condition is that elasticity of substitution ε and the multiplier dy /dg are fairly stable when tightness varies a lot. Indeed, ε and dy /dg are evaluated at [(G/C),x ] in (7) but at [G/C,x] in (11). If these two statistics varied a lot with tightness, the right-hand sides of (7) and (11) would be very different. We know little about ε and its possible variations over the business cycle. On the other hand, there is growing evidence that multipliers are higher when the unemployment rate is higher. 20 Of course, the variations of dy /dg when x varies only have a second-order effect in formula (7); but these second-order effects could be large is tightness drifts far from efficiency. 21 To gauge the quality of formula (7) when dy /dg responds to x, Section 5 develops a specific model in which dy /dg is countercyclical and investigates numerically whether formula (7) provides a good approximation to the exact formula (5) An Explicit Formula in Estimable Statistics While formula (7) is useful for certain applications, we cannot use the formula to answer the following question: if the tightness is 50% above its efficient level and government purchases are at the Samuelson level, what should be the optimal increase in government purchases? This is because the formula describes the optimal policy implicitly. 22 This is a typical limitation of sufficient-statistics optimal policy formulas, and a typical criticism addressed to the sufficientstatistics approach [Chetty, 2009]. Here we develop an explicit sufficient-statistics formula that can be used to address this question. We assume that the tightness is initially at an inefficient level x 0 x. As government purchases change, tightness endogenously responds. Once we have described the endogenous response, we obtain the following explicit formula: PROPOSITION 4. Assume that the economy is at an equilibrium [(G/C),x 0 ], where the tightness 20 See for instance Auerbach and Gorodnichenko [2012, 2013]. 21 By second-order effect, we mean that accounting for the variation of dy /dg when x deviates from x would only add a term that is O((x x ) 2 ) to formula (7). 22 The ratio G/C is implicitly defined by (7) because the right-hand side of (7) is endogenous to G/C. 21

22 x 0 x is inefficient. Then optimal government purchases are approximately given by where G/C (G/C) (G/C) ε m 1 + a ε m 2 x0 x x, (12) a (G/Y ) (1 G/Y ), (13) (1 η) u m = (dy /dg)/(1 (G/Y ) (dy /dg)) and (G/C), ε, x, and 1 η are defined above and listed in Table 2. The statistics ε, m, and a are evaluated at [(G/C),x ]. Furthermore, the equilibrium level of tightness once optimal government purchases are in place is approximately given by x x a ε m 2 (x 0 x ). (14) The two approximations are valid up to a remainder that is O((x 0 x ) 2 + (G/C (G/C) ) 2 ). The complete proof is in Appendix C. It is easy to derive the result heuristically starting from the implicit formula (7) and recognizing that x x 0 +α (G/C (G/C) ) where α is the marginal effect of G/C on x. Plugging this expression into (7) and re-arranging, we obtain G/C (G/C) (G/C) ε m 1 + ε m α (G/C) /x x0 x x. The proof shows that α m a x /(G/C) which immediately yields (12). 23 Taking the ratio of (7) and (12) immediately implies that x x (x 0 x )/(1 + a ε m), which proves (14). Formula (12) links the deviation of the optimal government purchases-output ratio from the Samuelson ratio to the initial deviation of tightness from its efficient level. The formula can be directly applied by policymakers to determine the optimal response of government purchases to a shock that leads to a departure of tightness from its efficient level. Since formula (12) builds on formula (7), formula (12) requires the same conditions as formula (7) to be accurate when tightness moves far from its efficient level. Section 5 uses numerical simulations to investigate the accuracy of the formula. 23 As α captures the effect of G/C on x, it is not surprising that it is proportional to the multiplier m. 22